arXiv subjects

Yoshiyuki Kitaoka

Publications and source records attributed to Yoshiyuki Kitaoka.

4 recordsLinked to original sources

arXiv2024

Conjectures on the distribution of roots modulo a prime of a polynomial

For a given monic integral polynomial $f(x)$ of degree $n$, we define local roots $r_i$ of $f(x)$ for a completely decomposable prime $p$ by $r_i \in \mathbb{Z}$, $f(r_i) \equiv 0 \bmod p$ and $0 \le r_1 \le r_2 \le \dots \le r_n < p$. With numerical data, we propose a conjecture on the distribution of $(r_1/p,\dots,r_n/p)$, which is a new kind of equi-distribution, and a conjecture of the distribution of $(r_1,\dots,r_n)$ which satisfies $r_i \equiv R_i \bmod L$ for given natural numbers $L,R_1,\dots,R_n$, which is nothing but Dirichlet's theorem on an arithmetic progression in the case $n = 1$.

math.NT↗

arXiv2018

Linear relations among roots of a polynomial

Let $f(x)$ be a monic polynomial over $\mathbb{Q}$ with complex roots $α_1,\dots,α_n$. Linear relations among them and $1$ over $\mathbb{Q}$ play an important role when we study the distribution of roots modulo a prime. We study linear relations group-theoretically. If $\mathbb{Q}(α_1)$ is an abelian extension, we can understand the structure completely.

math.NT↗

arXiv2017

Statistical distribution of roots of a polynomial modulo primes

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le r_n<p$ and propose several conjectures on the distribution of an integer $\lceil \sum_{i\in S} r_i/p\rceil$ for a subset $S$ of $\{1,\dots,n\}$ when $p\to\infty$.

math.NT↗

arXiv2017

Behaviour of the least root modulo a prime of a polynomial

For a polynomial $f(x)\in\mathbb Z[x]$ without non-trivial linear relations among roots, we propose a conjecture on the distribution of the least root $r_p$ ($r_p\in\mathbb Z,\,0\le r_p<p)$ of $f(x)\equiv0\bmod p$ where $p$ runs over the set of primes such that $f(x)$ modulo $p$ is fully splitting.

math.NT↗